This is not a simple problem.
. . I hope I got it right . . .
A manufacturer of merry-go-rounds uses 8 identical wooden horses,
4 identical plastic motorbikes and 2 different miniature car.
They are all equally connected around the rim of a circular moving base.
Establish how many different arrangements there can be
if the two cars are not to be placed in consecutive positions.
There are 8 identical horses, 4 identical bikes and 2 different cars.
The objects are: .
If they were arranged in a row, there are: . arrangements.
. . Since they are in a circle, there are: . arrangements.
Now we'll find the number of ways the two cars are together.
Duct-tape the two cars together.
Then we have 13 "objects" to arrange: .
. . In a row, there are: . arrangements.
But the two cars could be taped like this: .
. . which creates another 6435 arrangements, for a total of 12,870 ways.
But since they are arranged in a circle, there are: . ways.
Recap: .There are 6435 possible arrangements.
. . . . . .In 990 of them, the two cars are adjacent.
Therefore, there are: . ways in which the cars are not adjacent.